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Variational Problem in Euclidean Space With Density

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Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Variational Problem in Euclidean Space With Density Lakehal BELARBI1 and Mohamed BELKHELFA2 1Departement de Math´ematiques, Universit´e de Mostaganem B.P.227,27000,Mostaganem, Alg´erie. 2Laboratoire de Physique Quantique de la Mati`ere et Mod´elisations Math´ematiques (LPQ3M), Universit´e de Mascara B.P.305 , 29000,Route de Mamounia Mascara, Alg´erie. GEOMETRIC SCIENCE OF INFORMATION Paris-Ecole des Mines 28,29 and 30 August 2013 1 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Outline 1 Introduction : • What is a manifold with density. • Examples of a manifold with density. 2 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Outline 1 Introduction : • What is a manifold with density. • Examples of a manifold with density. 2 Preliminaries: 2 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Outline 1 Introduction : • What is a manifold with density. • Examples of a manifold with density. 2 Preliminaries: 3 Plateau’s problem in R3 with density. • Theorem. • Motivation. 2 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Outline 1 Introduction : • What is a manifold with density. • Examples of a manifold with density. 2 Preliminaries: 3 Plateau’s problem in R3 with density. • Theorem. • Motivation. 4 The Divergence operator in manifolds with density. 2 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References What is a manifold with density A manifold with density is a Riemannian manifold Mn with positive density function eϕ used to weight volume and hyperarea (and sometimes lower-dimensional area and length).In terms of underlying Riemannian volume dV0 and area dA0 , the new weighted volume and area are given by dV = eϕ .dV0, dA = eϕ .dA0. 3 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Examples of a manifold with density One of the first examples of a manifold with density appeared in the realm of probability and statistics, Euclidean space with the Gaussian density e−π|x| (see ([13]) for a detailed exposition in the context of isoperimetric problems). 4 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References For reasons coming from the study of diffusion processes,Bakry and ´Emery ([1]) defined a generalization of the Ricci tensor of Riemannian manifold Mn with density eϕ (or the ∞−Bakry-´Emery-Ricci tensor) by Ric∞ ϕ = Ric − Hessϕ, (1) where Ric denotes the Ricci curvature of Mn and Hessϕ the Hessian of ϕ. 5 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References By Perelman in ([11],1.3,p.6),in a Riemannian manifold Mn with density eϕ in order for the Lichnerovicz formula to hold, the corresponding ϕ−scalar curvature is given by S∞ ϕ = S − 2∆ϕ− | ϕ |2 , (2) where S denotes the scalar curvature of Mn.Note that this is different than taking the trace of Ric∞ ϕ which is S − ∆ϕ. 6 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Following Gromov ([6],p.213), the natural generalization of the mean curvature of hypersurfaces on a manifold with density eϕ is given by Hϕ = H − 1 n − 1 d ϕ dN , (3) where H is the Riemannian mean curvature and N is the unit normal vector field of hypersurface . 7 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References For a 2-dimensional smooth manifold with density eϕ , Corwin et al.([5],p.6) define a generalized Gauss curvature Gϕ = G − ∆ϕ. (4) and obtain a generalization of the Gauss-Bonnet formula for a smooth disc D: D Gϕ + ∂D κϕ = 2π, (5) where κϕ is the inward one-dimensional generalized mean curvature as (1.3) and the integrals are with respect to unweighted Riemannian area and arclength ([9],p.181). 8 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Bayle ([2]) has derived the first and second variation formulae for the weighted volume functional (see also [9],[10],[13]).From the first variation formula, it can be shown that an immersed submanifold Nn−1 in Mn is minimal if and only if the generalized mean curvature Hϕ vanishes (Hϕ = 0). Doan The Hieu and Nguyen Minh Hoang ([8]) classified ruled minimal surfaces in R3 with density Ψ = ez. 9 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References In ([4]) , we have previously written the equation of minimal surfaces in R3 with linear density Ψ = eϕ (in the case ϕ(x, y, z) = x, ϕ(x, y, z) = y and ϕ(x, y, z) = z), and we gave some solutions of the equation of minimal graphs in R3 with linear density Ψ = eϕ. In ([3]),we gave a description of ruled minimal surfaces by geodesics straight lines in Heisenberg space H3 with linear density Ψ = eϕ = eαx+βy+γz,where (α, β, γ) ∈ R3 − {(0, 0, 0)} (in particular ϕ(x, y, z) = αx and ϕ(x, y, z) = βy), and we gave the ∞−Bakry-´Emery Ricci curvature tensor and the ϕ−scalar curvature of Heisenberg space H3 with radial density e−aρ2+c,where ρ = x2 + y2 + z2. 10 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. We deal with two-dimensional surfaces in Euclidean 3-space.We assume that the surface is given parametrically by X : U ⊆ R2 → R3. We denote the parameters by u and v. We denote the partial derivatives with respect to u and v by the corresponding subscripts. The normal vector N to the surface at a given point is defined by N = Xu ∧ Xv Xu ∧ Xv . 11 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References The first fundamental form of the surface is the metric that is induced on the tangent space at each point of the surface.The (u, v) coordinates define a basis for the tangent space.This basis consists of the vectors Xu and Xv .In this basis the matrix of the first fundamental form is E F F G , where E = Xu.Xu, F = Xu.Xv , and G = Xv .Xv . In this basis, the second fundamental form of the surface is given by the matrix : L M M N , where L = −Xu.Nu, M = −Xu.Nv , and N = −Xv .Nv . 12 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Definition ([12]) The area AX(R) of the part X(R) of a surface patch X : U ⊆ R2 → R3 corresponding to a region R ⊆ U is AX(R) = R Xu ∧ Xv dudv. and Xu ∧ Xv = (EG − F2 ) 1 2 . 13 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References We shall now study a family of surface St parameterized by Xt : U → R3 in R3 with density eϕt ,where U is an subset of R2 independent of t,and t lies in some open interval ] − , [,for some > 0. Let S = S0 and eϕ0 = eϕ .The family is required to be smooth, in the sense that the map (u, v, t) → Xt(u, v) from the open subset {(u, v, t)/(u, v) ∈ U, t ∈] − , [} of R3 to R3 is smooth. 14 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References We shall now study a family of surface St parameterized by Xt : U → R3 in R3 with density eϕt ,where U is an subset of R2 independent of t,and t lies in some open interval ] − , [,for some > 0. Let S = S0 and eϕ0 = eϕ .The family is required to be smooth, in the sense that the map (u, v, t) → Xt(u, v) from the open subset {(u, v, t)/(u, v) ∈ U, t ∈] − , [} of R3 to R3 is smooth. The surface variation of the family is the function η : U → R3 given by η = ∂Xt ∂t /t=0, Let γ be a simple closed curve that is contained,along with its interior int(γ),in U. Then γ corresponds to a closed curve γt = Xt ◦ γ in the surface St, and we define the ϕt−area function Aϕt (t) in R3 with density eϕt to be the area of the surface St inside γt in R3 with density eϕt : Aϕt (t) = int(γ) eϕt dAXt . 14 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Theorem Theorem With the above notation, assume that the surface variation ηt vanishes along the boundary curve γ.Then, ∂Aϕt (t) ∂t /t=0 = Aϕ(0) = −2 int(γ) Hϕ.η.N.eϕ .(EG − F2 ) 1 2 dudv, (6) where Hϕ = H − 1 2 ϕ.N is the ϕ−mean curvature of S in R3 with density eϕ , E, F and G are the coefficients of its first fundamental form,and N is the standard unit normal of S. 15 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Motavation If S in R3 with density eϕ has the smallest ϕ−area among all surfaces in R3 with density eϕ with the given boundary curve γ, then Aϕ must have an absolute minimum at t = 0, so Aϕ(0) = 0 for all smooth families of surfaces as above. This means that the integral in Eq.(6) must vanish for all smooth functions ζ = η.N : U → R. This can happen only if the term that multiplies ζ in the integrand vanishes,in other words only ifHϕ = 0. This suggests the following definition. 16 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Definition A minimal surface in R3 with density eϕ is a surface whose ϕ−mean curvature is zero everywhere. 17 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Proposition The minimal equation of surface S : z = f (x, y) in R3 with linear density ex given by the parametrization: X : (x, y) → (x, y, f (x, y)) , where (x, y) ∈ R2 is 1 + ∂f ∂x 2 ∂2f ∂y2 + ∂f ∂x + 1 + ∂f ∂y 2 ∂2f ∂x2 + ∂f ∂x −2 ∂f ∂x . ∂f ∂y . ∂2f ∂x∂y − ∂f ∂x = 0. 18 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Example The surface S in R3 with linear density ex defined by the parametrization : X : (x, y) → x, y, − a2 √ 1 + a2 arcsin(βe − 1+a2 a2 x ) + ay + b + γ , where (x, y) ∈ R2 , a, b, β ∈ R∗ is minimal. 19 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Let (Mn, g) be a Riemannian manifold equipped with the Riemannian metric g.For any smooth function f on M, the gradient f is a vector field on M, which is locally coordinates x1, x2....., xn has the form ( f )i = gij ∂f ∂xj , where summation is assumed over repeated indices. For any smooth vector field F on M, the divergence divF is a scalar function on R, which is given in local coordinates by divF = 1 detgij ∂ ∂xi ( detgij Fi ) Let ν be the Riemannian volume on M, that is ν = detgij dx1.....dxn. By the divergence theorem, for any smooth function f and a smooth vector field F, such that either f or F has compact support, M fdivFdν = − M < f , F > dν. (7) 20 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References where < ., . >= g(., .). In particular, if F = ψ for a function ψ then we obtain M fdiv ψdν = − M < f , ψ > dν. (8) provided one of the functions f , ψ has compact support. The operator ∆ := div ◦ is called the Laplace (or Laplace-Beltrami ) operator of the Riemannian manifold M. From (8) we obtain the Green formulas M f ∆ψdν = − M < f , ψ > dν = M ψ∆fdν. (9) 21 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Let now µ be another measure on M defined by dµ = eϕ dν where ϕ is a smooth function on M. A triple (Mn, g, µ) is called a weighted manifold or manifold with density.The associative divergence divµ is defined by divµF = 1 eϕ detgij ∂ ∂xi (eϕ detgij Fi ), and the Laplace-Beltrami operator ∆µ of (Mn, g, µ) is defined by ∆µ. := divµ ◦ . = 1 eϕ div(eϕ .) = ∆. + ϕ .. (10) It is easy to see that the Green formulas hold with respect to the measure µ, that is, M f ∆µψdµ = − M < f , ψ > dµ = M ψ∆µfdµ. (11) provided f or ψ belongs to C∞ 0 (M). 22 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Theorem Let S a surface in M3 with density Ψ = eϕ, we have divϕN = −2Hϕ. (12) where Hϕ is the ϕ−mean curvature of a surface S and N is the unit normal vector field of a surface S. 23 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References Proof. by definition we have divϕN = 1 eϕ div(eϕN).+ < ϕ, N > = divN + N. ϕ = ∑i=2 i=1 < ei N, ei > +N. ϕ = ∑i=2 i=1( ei < ei , N > − < ei ei , N >) + N. ϕ = −2 < HN, N > +N. ϕ = −2(H − 1 2 ϕ.N) = −2Hϕ. where we have used that < ei , N >= 0 and the definition of the mean curvature vector. 24 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References D.Bakry,M.´Emery.Diffusions hypercontractives. S´eminaire de Probabilit´es,XIX, 1123 (1983/1984,1985), 177-206. V.Bayle,Propri´et´es de concavit´e du profil isop´erim´etrique et applications.graduate thesis,Institut Fourier,Univ.Joseph-Fourier,Grenoble I ,(2004). L.Belarbi,M.Belkhelfa.Heisenberg space with density,( submetted) . L.Belarbi,M.Belkhelfa.Surfaces in R3 with density,i-manager’s Journal on Mathematics,Vol. 1 .No. 1.(2012),34-48. I.Corwin,N.Hoffman,S.Hurder,V.Sesum,and Y.Xu,Differential geometry of manifolds with density,Rose-Hulman Und.Math.J.,7(1) (2006). 25 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References M.Gromov, Isoperimetry of waists and concentration of maps,Geom.Func.Anal 13(2003),285-215. J.Lott,C.Villani,Ricci curvature metric-measure space via optimal transport.Ann Math,169(3)(2009),903-991. N.Minh,D.T.Hieu,Ruled minimal surfaces in R3 with density ez,Pacific J. Math. 243no. 2 (2009), 277–285. F.Morgan,Geometric measure theory,A Beginer’s Guide,fourth edition,Academic Press.(2009). 26 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References F.Morgan,Manifolds with density,Notices Amer.Math.Soc.,52 (2005),853-858. G.Ya.Perelman,The entropy formula for the Ricci flow and its geometric applications,preprint,http://www.arxiv.org/abs/math.DG/0211159. (2002). A.Pressley.Elementary Differential Geometry,Second Edition,Springer. (2010). C.Rosales,A.Ca˜nete,V.Bayle,F.Morgan.On the isoperimetric problem in Euclidean space with density.Cal.Var.Partial Differential Equations.31(1) (2008) 27-46. 27 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density Introduction Preliminaries Plateau’s problem in R3 with density The Divergence operator in manifolds with density References THANK YOU FOR YOUR ATTENTION 28 / 28 Lakehal BELARBI and Mohamed BELKHELFA Variational Problem in Euclidean Space With Density